Combine antecedents into a single bi-conditional. This inference,
reminiscent of ja155, is reversible: The hypotheses can be deduced
from
the conclusion alone (see pm5.1im230 and pm5.21im339). (Contributed by
Wolf Lammen, 13-May-2013.)

Associative law for the biconditional. An axiom of system DS in Vladimir
Lifschitz, "On calculational proofs", Annals of Pure and Applied
Logic,
113:207-224, 2002,
http://www.cs.utexas.edu/users/ai-lab/pub-view.php?PubID=26805.
Interestingly, this law was not included in Principia Mathematica
but
was apparently first noted by Jan Lukasiewicz circa 1923. (Contributed by
NM, 8-Jan-2005.) (Proof shortened by Juha Arpiainen, 19-Jan-2006.)
(Proof shortened by Wolf Lammen, 21-Sep-2013.)

Simplify an implication between two implications when the antecedent of
the first is a consequence of the antecedent of the second. The reverse
form is useful in producing the successor step in induction proofs.
(Contributed by Paul Chapman, 22-Jun-2011.) (Proof shortened by Wolf
Lammen, 14-Sep-2013.)

Define disjunction (logical 'or'). Definition of [Margaris] p. 49. When
the left operand, right operand, or both are true, the result is true;
when both sides are false, the result is false. For example, it is true
that (ex-or21591). After we define the constant
true (df-tru1325) and the constant false (df-fal1326), we
will be able to prove these truth table values:
(truortru1346),
(truorfal1347), (falortru1348), and
(falorfal1349).

This is our first use of the biconditional connective in a definition; we
use the biconditional connective in place of the traditional
"<=def=>",
which means the same thing, except that we can manipulate the
biconditional connective directly in proofs rather than having to rely on
an informal definition substitution rule. Note that if we mechanically
substitute for , we end up
with an
instance of previously proved theorem biid228.
This is the justification
for the definition, along with the fact that it introduces a new symbol
. Contrast
with (df-an361), (wi4),
(df-nan1294), and (df-xor1311) . (Contributed by NM,
5-Aug-1993.)

Define conjunction (logical 'and'). Definition of [Margaris] p. 49. When
both the left and right operand are true, the result is true; when either
is false, the result is false. For example, it is true that
. After we define the
constant true
(df-tru1325) and the constant false (df-fal1326), we will be able
to prove these truth table values:
(truantru1342), (truanfal1343),
(falantru1344), and
(falanfal1345).

Deduction eliminating disjunct. Notational convention: We sometimes
suffix with "s" the label of an inference that manipulates an
antecedent, leaving the consequent unchanged. The "s" means
that the
inference eliminates the need for a syllogism (syl16)
-type inference
in a proof. (Contributed by NM, 21-Jun-1994.)